Tag: dissecting polygons

Posted in GeoGebra worksheets, Geometry

How to scaffold problem solving in geometry

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Scaffolding is a metaphor for describing a type of facilitating a teacher does to support students’ own making sense of things. It is usually in the form of questions or additional information. In scaffolding learning, we should be careful not to reduce the learning by rote. In the case of problem solving for example, the scaffolds provided should not reduce the problem solving activity into one where students follow procedures disguised as scaffolds. So how much scaffolding should we provide? Where do we stop? Let us consider this problem:

ABCD is a square. E is the midpoint of CD. AE intersects the diagonal BD at F.

  1. List down the polygons formed by segments BD and AE in the square.
  2. How many percent of the area of square ABCD is the area of each of the polygons formed?

Students will have no problem with #1. In #2, I’m sure majority if not all will be able to compare the area of triangles ABD, BCD, AED and quadrilateral ABCE to the area of the square. The tough portion is the area of the other polygons – ABF, AFD, FED, and BCEF.

In a problem solving lesson, it is important to allow the learners to do as much as they can on their own first, and then to intervene and provide assistance only when it is needed. In problems involving geometry, the students difficulty is in visualizing the relationships among shapes. So the scaffolding should be in helping students to visualize the shapes (I actually included #1 as initial help already) but we should never tell the students the relationships among the geometric figures. I created a GeoGebra worksheet to show the possible scaffolding that can be provided so students can answer question #2. Click here to to take you to the GeoGebra worksheet.

 

Extension of the problem: What if E is 1/4 of its way from C to D? How many percent of the square will be the area of the three triangles and the quadrilateral? How about 1/3? 2/3? Can it be generalized?

Please share with other teachers. I will appreciate feedback so I can improve the activity. Thank you.

More Geometry Problems:

  1. The Humongous Book of Geometry Problems: Translated for People Who Don’t Speak Math
  2. Challenging Problems in Geometry

 

Posted in Trigonometry

Trigonometry – why study triangles

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Why study trigonometry?

We study Trigonometry because it is useful. Its earliest and simplest use is to find the missing part of a triangle. But mathematicians do not just study something because it is useful. More often, they study something because it is fascinating. This fascination with triangles especially in the measure of it sides and angles has developed into a coherent piece of mathematical knowledge we now call trigonometry.

Why not quadrinometry?

What is so special about triangles? Why did mathematicians created a branch of mathematics devoted to the study of it? Why not quadrinometry? Quadrilaterals, by its variety are far more interesting. Not only that, each piece of shape is related to another piece. If you know quadrilateral, you’ll know about the rest of the quadrilaterals. But this is also precisely the reason why we study trigonometry, why we study triangles. If you know it, you’ll know about any polygon not just quadrilaterals. Any polygon can be dissected into pieces of triangles! Try dissecting any of these shapes:

What’s with right triangles?

There are different kinds or shapes of triangles. In terms of angles we have equiangular, acute, obtuse, and right triangles. Why is it that we devote so much time studying about right triangles in trigonometry? Try dissecting the other triangles and you’ll know why if you know about right triangles, you’ll know about the other triangles!.

 

 

 

 

 

 

 

 

 

 

 

Here’s a bonus reason: when you study triangles, you don’t need to deal with nonconvex ones!
Click Teaching trigonometry through problem solving for a sample lesson on teaching trigonometry in presentation format.

I created a worksheet for the activities on classifying and dissecting polygons. Click the link.